We need to make a string of size $n$. Each character of the string is either ‘R’, ‘B’ or ‘G’. In the final string there needs to be at least r number of ‘R’, at least b number of ‘B’ and at least g number of ‘G’ (such that r + g + b <= n).
How to find the number of such constrained strings?
Let we count for first the number $N_{x,y,z}$ of constrained strings in which there are $x\geq r$ characters 'R', $y \geq g$ characters 'G' and $z\geq b$ characters 'B', with $x+y+z=n$. We have: $$ N_{x,y,z} = \binom{n}{x}\binom{n-x}{y}\binom{n-x-y}{z}=\frac{n!}{x!y!z!},$$ hence the total number of constrained strings is given by: $$n!\sum_{\substack{x\geq r,y\geq g,z\geq b \\ x+y+z=n}}\frac{1}{x!y!z!}.$$